Let us first construct \(\omega _X\)  by glueing together the analytifications of the dualizing complexes coming from Theorem REF . Let \(U=\mathop {\mathrm {Spa}}A\)  be any affinoid rigid space, and set \(\mathcal {U}=\mathop {\mathrm {Spec}}A\) . Then \(\mathcal {U}\)  satisfies the hypotheses of Theorem REF . Let \(\omega _{\mathcal {U}} \in D^b_c(\mathcal {U},\mathbf {Z}/n)\)  be the dualizing complex provided by that theorem.
By Propositions REF  and REF , we have an equivalence \((-)^{an}: D^b_c(\mathcal {U},\mathbf {Z}/n) \rightarrow D^b_{zc}(U,\mathbf {Z}/n)\)  compatible with \(\mathop {R{H}\mathrm {om}}\) s. We now define \(\omega _U := (\omega _{\mathcal {U}})^{an}\) , so \((\mathbf {D}_{\mathcal {U}}{F})^{an} \cong \mathbf {D}_U({F}^{an})\)  for every \({F} \in D^b_c(\mathcal {U},\mathbf {Z}/n)\) . Since the dualizing functor \(\mathbf {D}_{\mathcal {U}}\)  induces a contravariant self-equivalence on \(D^b_c(\mathcal {U},\mathbf {Z}/n)\)  which satisfies biduality, and \((-)^{an}\)  is an equivalence of categories, we now conclude the analogous results for \(\mathbf {D}_U\) .
To construct \(\omega _X\)  over an arbitrary rigid space \(X\) , we apply Lemma REF , taking \(\mathcal {B}\)  to be the collection of affinoid opens \(U \subset X\) , and letting \(K_U = \omega _U\)  as constructed in the previous paragraph. Note that the full faithfulness in Proposition REF  and Theorem REF  show that \(\mathbf {Z}/n \cong \mathop {R{H}\mathrm {om}}(\omega _U,\omega _U)\)  for all \(U \in \mathcal {B}\) , so all negative self-exts vanish. Moreover, for any inclusion of open affinoid subsets \(g: V \subset U\) , there is a natural isomorphism \(g^{\ast } \omega _U \cong \omega _V\)  compatible with compositions by Lemma REF  below (applied with \(d=0\) ). The gluing lemma now applies, and gives a unique \(\omega _{X} \in D(X,\mathbf {Z}/n)\)  equipped with a transitive system of isomorphisms \(\omega _X|_U \simeq \omega _U\)  for all open affinoids \(U\subset X\) .
We now prove this construction has all the required properties.

By construction, the complex \(\omega _X\)  is characterized by the properties demanded in the first sentence of (1); these also characterize \(\omega _X\)  uniquely (up to unique isomorphism) by the uniqueness assertions in Theorem REF  and Lemma REF  through the equivalence in Proposition REF . It is also clear from the construction and Remark REF  that \(\omega _X|_{X^{sm}} \cong \mathbf {Z}/n[2d](d)\) , where \(d\)  is the dimension. For the compatibility with \(Ri^!\)  for finite morphisms, we reduce to the affinoid case following our construction, and use compatibility of the analytification functor with \(Ri^!\)  (Proposition REF ) to reduce to the corresponding result in algebraic geometry ([1]}). The compatibility with restriction to open subsets is clear from our construction. As these properties guarantee uniqueness of \(\omega _X\)  up to isomorphism, we are done with (1).

The identification with \(R\pi _X^!(\mathbf {Z}/n)\)  can be checked locally, where it follows by factoring \(\pi _X\)  as a closed immersion followed by a smooth map and using the results on Poincare duality proved in Huber's book.

This follows from the corresponding assertion in the affinoid case (which was explained above whilst constructing \(\omega _X\) ) as the property of being Zariski-constructible is local in the analytic topology (Theorem REF ).

This is a formal argument given duality and known adjunctions. For compatibility with \(i_*\) , fix \(G \in D^{(b)}_{zc}(Z,\mathbf {Z}/n)\) . Then we have isomorphisms
\(i_* \mathbf {D}_Z(G) =& i_* \mathop {R{H}\mathrm {om}}(\mathbf {Z}/n,\mathbf {D}_Z(G)) \\=& i_* \mathop {R{H}\mathrm {om}}(G, \omega _Z) \\=& i_* \mathop {R{H}\mathrm {om}}(G, i^! \omega _X) \\=& \mathop {R{H}\mathrm {om}}(i_* G, \omega _X) \\=& \mathbf {D}_X(i_* G).\) 

where the second isomorphism is by biduality, the third by \(\omega _Z = i^! \omega _X\) , and the fourth by the defining property of \(i^!\) . Comparing the first and last term gives \(i_* \mathbf {D}_Z = \mathbf {D}_X i_*\) .
For compatibility with \(i^!\) , fix additionally \(F \in D^{(b)}_{zc}(X,\mathbf {Z}/n)\) . Then we have bifunctorial isomorphisms
\(\mathrm {RHom}(i^* \mathbf {D}_X(F), G) =& \mathrm {RHom}(\mathbf {D}_X(F), i_* G)\\=& \mathrm {RHom}(\mathbf {D}_X(i_* G), F) \\=& \mathrm {RHom}(i_* \mathbf {D}_Z(G),F) \\=& \mathrm {RHom}(\mathbf {D}_Z(G), Ri^! F) \\=& \mathrm {RHom}(\mathbf {D}_Z(Ri^! F),G),\) 

where first equality is by adjunction for \((i^*,i_*)\) , the second and last by duality, the third by the equality \(i_* \mathbf {D}_Z = \mathbf {D}_X i_*\)  we just showed, and the fourth by adjunction for \((i_*,i^!)\) .

The compatibility with \(j^*\)  is built into the construction. The rest is again a formal argument using duality and known adjunctions. For \(F \in D^{(b)}_{zc}(U,\mathbf {Z}/n)\)  and \(G \in D^{(b)}_{zc}(X,\mathbf {Z}/n)\) , we have
\(\mathrm {RHom}(F, Rj_* \mathbf {D}_U(G)) =& \mathrm {RHom}(j^* F, \mathbf {D}_U(G)) \\=& \mathrm {RHom}(G, \mathbf {D}_U (j^* F)) \\=& \mathrm {RHom}(G, j^* \mathbf {D}_X(F)) \\=& \mathrm {RHom}(j_! G, \mathbf {D}_X(F)) \\=& \mathrm {RHom}(F, \mathbf {D}_X(j_! G)),\) 

with first equality by the adjunction \((j^*,Rj_*)\) , the second and last by duality, third by \(j^* \mathbf {D}_X = \mathbf {D}_U j^*\) , and the fourth by the adjunction \((j_!, j^*)\) .

By our construction of dualizing complexes, it suffices to construct a natural system of such isomorphisms over affinoids. Thus, assume \(X=\mathrm {Spa}(A)\)  is affinoid with base change \(X_L = \mathrm {Spa}(A_L)\) . Write \(\mathcal {X}\)  and \(\mathcal {X}_L\)  for the natural algebraization, and let \(f:\mathcal {X}_L \rightarrow \mathcal {X}\)  be the natural map. It is enough to construct a natural isomorphism \(f^* \omega _{\mathcal {X}} \simeq \omega _{\mathcal {X}_L}\) . We claim this follows from [1]} (and uniqueness of potential dualizing complexes). To apply this lemma, we need to know that \(f\)  is a regular map, and that the dimension function \(y \mapsto \delta ^{\prime }(y) := \delta (f(y)) - \mathrm {codim}_{f^{-1}(f(y))}(y)\)  on \(\mathcal {X}_L\)  agrees with the standard dimension function \(y \mapsto \dim (\overline{\lbrace y\rbrace })\) . The regularity of \(f\)  is discussed in the last paragraph of the proof of Proposition REF  below, and essentially comes from [3]}. To obtain agreement of the dimension functions, it suffices to know that \(\delta ^{\prime }(y) = 0\)  for a closed point \(y\)  (as any point on \(\mathcal {X}_L\)  is linked to a closed point by a finite chain of immediate specializations). Thus, we must check that \(\dim (\overline{\lbrace f(y)\rbrace }) = \mathrm {codim}_{f^{-1}(f(y))}(y)\) . Since \(y\)  is a closed point, this follows from [4]} applied to the affinoid algebra of functions on integral scheme \(\overline{\lbrace y\rbrace } \subset \mathcal {X}\) .

To deduce this from our construction, it suffices to show the following: if \(\mathcal {X} = \mathrm {Spec}(A)\)  is an affine finite-type \(K\) -scheme and \(U = \mathrm {Spa}(B) \subset X = \mathcal {X}^{an}\)  is an open affinoid with natural map \(\nu :U \rightarrow \mathcal {X}\) , then there is a natural isomorphism \(\nu ^* \omega _{\mathcal {X}} \simeq \omega _U\) . This follows by Lemma REF  below applied to the map \(\mathrm {Spec}(B) \rightarrow \mathrm {Spec}(A)\)  with \(d=0\) .

